Raffles Institution 04 2024 Y3 SUPP WS Quadratic Functions

Page 1 of 5 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2024 YEAR 3 RP MATHEMATICS TOPIC 4: QUADRATIC FUNCTIONS (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 3 ( ) Date: 1 2022/Y3RP/M2/T1/Q1 (i) Find the coordinates of the turning point of the curve ( )( )1 5 11 12yx x=+− . [2] (ii) Hence sketch the graph of ( )( )1 5 11 12yx x=+− for 31 x−≤ ≤ , showing all the critical points clearly. [3] [Ans: (i) 32, 655 − ] 2 2022/Y3RP/M2/T1/Q2 The equation of a curve is 2 933y ax x a= ++− , where a is a constant and 0a≠ . (a) Show that the line ( )3y xa= + intersects the curve at two distinct points for all real values of a ( 0a≠ ). [5] (b) Find the range of values of a for which the curve lies completely below the line 3 4y=− . [5] [Ans: (b) 12 4a<− ] 3 2021/Y3RP/M2/T1/Q1 Find the range of values of a such that 2 35x ax a++− is always positive. [3] [Ans: 2 10a<< ] 4 2021/Y3RP/M2/T1/Q2 Find the range of values of k for which the line y kx k+= and the curve ( ) 2 2 1 10y kx k x k=+++ do not meet. [4] [Ans: 11 or 93kk<− > ] 5 2021/Y3RP/M2/T1/Q3 Show that ( ) ( ) ( ) 22 2 3 13 0mx mx m− + + +− = has real and distinct roots for all real values of m such that 2m≠ . [4] 6 2021/Y3RP/M2/T1/Q4 (i) The equation of a curve is ( )( )2 13yx x=+− . Find the maximum value of y and the corresponding value of x. [2] [Ans: (i) 49 5Maximum value of when 84yx= = ]

Page 2 of 5 7 2020/Y3RP/M2/T1/Q1 Find the range of values of p for which the curve ( ) ( ) 2 3 41y x p px=−− − has a positive minimum y value. [4] [Ans: 33 44 p−<< ] 8 2020/Y3RP/M2/T1/Q2 Show that the line ( )43 1yx m= +− cuts the curve ( ) 2 21 2yx m x= + −+ at two distinct points for all real values of m . [5] 9 2019/Y3RP/M2/T2/Q2 Find the range of values of m for which the equation 2 45 4x x mx− += + has no real roots. Hence, determine the number of points of intersection between the curve ( ) 2 2yx= − and the line 23yx= + . [5] [Ans: 62 m− < <− ; 2 points of intersection] 10 2019/Y3RP/M2/T2/Q4 Find the range of values of p for which ( ) 268p xp x− + >− for all real values of x.[5] [Ans: p > 8] 11 2018/Y3RP/M2/T1/Q1 Find the range of values of k such that the equation 2 2 4 4 ( 2)x x kx− −= + has no real roots. [4] [Ans: 112 < 22 k− <− ] 12 2018/Y3RP/M2/T1/Q2 Show that 2232k kx x−++ is positive for all real values of x. [3] 13 2018/Y3RP/M2/T1/Q3 Find the range of values of the constant p for which the line 41yx= − intersects the curve 282 1py x px−= + at two distinct points. [4] [Ans: 1 or 8 and 0pp p<> ≠ ] 14 2017/Y3RP/T2/Q1 Find the range of values of k for which the equation 22 10x kx x k+ +− += has real roots. [3] [Ans: 31 or 5kk≤− ≥ ] 15 2017/Y3RP/T2/Q2 (i) Given that 2 8ax x c−+ is always negative, what conditions must apply to the constants a and c? [3] (ii) Give an example of values of a a nd c which satisfy the conditions found in part (i). [1] [Ans: (i) a < 0 and ac > 16] 16 2017/Y3RP/T2/Q3 Show that the line 2y kx k= + intersects the curve ( ) 221y k x kx=+ +− at two distinct points for all real values of k, where 2k ≠− . [5] 17 2016/Y3RP/T2/Q1 Solve the inequality ( )( ) 21 5 2 ( 1)x xx− − <− . [2] [Ans: x < 1 or x > 2]

Page 3 of 5 18 2016/Y3RP/T2/Q3 Show that the equation 3 ( )(2 ) 0x ax x a+ + −+= has real and distinct roots for all real values of a. [4] 19 2016/Y3RP/T2/Q6 (modified) The solution of the inequality 220x px q+ +< , where p and q are constants, is 36 x−< < . (i) Find the value of p and of q. [2] (ii) Find the coordinates of the turning point of the curve 22y x px q= ++ . [2] (iii) If the inequality 220x px q c+ ++< , where c is an integer, has no solut ion, state the minimum value of c. [1 ] [Ans: (i) p = ̶ 6, q = ̶ 36 (ii) 31, 4022  − (iii) 41] 20 2015/Y3RP/T2/Q1 Solve the inequality ( )( )3 7 5 17xx+ −< . [2] [Ans: 22 or 5xx<− > ] 21 2015/Y3RP/T2/Q2 Calculate the range of values of c for which 2 139 2 4x xc− +> for all real values of x. [3] [Ans: c > 9] 22 2015/Y3RP/T2/Q3 Show that the line 26yx= − cuts the curve ( ) 2226y x k xk=+−− at two distinct points for all values of k. [5] 23 2015/Y3RP/T2/Q4 Find the turning point of the quadratic function ( )( )32 5y xx= +− . Hence sketch the graph of ( )( )32 5y xx= +− , showing the intercepts on the axes and the turning point clearly. [5] [Ans: 311 , 2148   ] 24 2014/Y3RP/T3/Q2 Find the least value of integer b for which 235x bx− +− is negative for all real values of x. [3] [Ans: −7] 25 2014/Y3RP/T3/Q3 Find the range of values of k , where 0k ≠ for which the equation ( ) ( ) 2 6 2 3 20kx k x k+ − + += has real and distinct roots. [4] [Ans: k < 1 or k > 6] 26 2014/Y3RP/T3/Q4 (a) By completing the square, express 24 42xx−− in the form ( ) 2 ax h k−+ , where a, h and k are constants. [2] (b) Sketch the graph of 24 42yx x= −− . [3] [Ans: (a) 2 143 2x −− ]

Page 4 of 5 27 2013/Y3RP/T3/Q3 Find the range of values of k for which the line 22yx k= − intersects the curve 32yx x= + at two distinct points, leaving your answers in surd form. [4] [Ans: 32 o r 32kk<− > ] 28 2011/Y3RP/T2/Q2 Find the range of m for which the line intersects the curve at two distinct points. [3] [Ans: m < −6 or m > −2] 29 2011/Y3RP/T2/Q4 Express in the form where a, b and c are constants. Sketch the curve, stating clearly on the sketch where the curve cuts the y-axis and where the minimum occurs. [5] [Ans: 2 5 1134 6 12x ++ ] 30 2009/Y3RP/T3/Q2 Given that q is a constant, show that 2(2 ) (2 )qx x x q++− is positive for all real values of x. [4] 31 2009/Y3RP/T3/Q3 (a) Sketch the graph of ( ) 2 4 2 12yx= −+ for 24 x−≤≤ in the space below, indicating clearly the x - and y-intercepts (if any), the turning point and the end points. [3] (b) Find the range of values of k for which the line 1y kx= + has two distinct points of intersection with the curve ( ) 2 4 1 12yx=−+ . [3] [Ans: (b) k < −2 or k >1] 32 2008/Y3RP/T2/Q2 (a) Find the range of values of k for which the equation 2 2 522x kx k x+ += − − has real roots. [4] (b) Show that 2(3 ) 4 (2 )qx x x q++ − is positive for all real values of x. [4] [Ans: (a) k ≤ −2 or k ≥ 2] 33 2008/Y3RP/T3/Q1 The graph of a quadratic function ()y fx= meets the x-axis at the points (–3, 0) and (1, 0). Given that the graph of the function also passes through the point ( –1, –20), find the quadratic function. [3] [Ans: ( )( ) 25 3 1 or 5 10 15y x x yx x=+ − =+− ] 34 2008/Y3RP/T3/Q4 Express 23 92y xx= −+− in the form ( ) 2 y ax h k= −+ where a, h and k are constants. Sketch the graph of 23 9 2 for 0 4y xx x= − + − ≤≤ . [5] [Ans: 2 3334 24x−−+ ] 35 2006/ Y3RP/T2/Q4

Page 5 of 5 Find the range of values of k for which the straight line 12y kx= − does not intersect the curve 2 4y x kx=−− , leaving your answer in surd form if necessary. [4] [Ans: 22 22 k− << ]